Please provide correct answer. Suppose / (x,y) = x\\y, P = (-4,4) and v = -3i -

Question

Please provide correct answer.

Suppose / (x,y) = xy, P = (-4,4) and v = -3i - 2j. Find the gradient of f. f = 1/y i+ -x/y lambda 2 j Note: Your answers should be expressions of x and y; e.g. "3x - 4y" Find the gradient of fat the point P. (f) (P) = 0.25 i+ 0.25 Note: Your answers should be numbers Find the directional derivative of fat P in the direction of v. Duf = Note: Your answer should be a number Find the maximum rate of change of fat P. Note: Your answer should be a number Find the (unit) direction vector in which the maximum rate of change occurs at P. u= i+ j Note: Your answers should be numbers

Explanation / Answer

f(x,y) = x/y

fx(x,y) = 1/y

fy(x,y) = -x/y^@

A . (1/y) i + (-x/y^2) j

B = 1/4 i + (4/4^2) j

= 0.25 i + 0.25 j

C .v = -3i - 2j

directional derivaitve of f at p in driection of V

0.25 i + 0.25 j . (-3i - 2j)

0.25*-3 - 0.25*2

= -5*0.25

= -1.25

D. max rate of change = sqrt( 1/16 + 1/6) = sqrt(2/16) = sqrt(1/8) = 1/2 sqrt(2)

E . unit vector is

1/4 / 1/ 2 sqrt(2) i + 1/4 / 1/ 2 sqrt(2) j

2 sqrt(2)/4 i + 2 sqrt(2)/4 j

= sqrt(2)/ i + sqrt(2)/ 2 j

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